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En mathématiques, l’escroquerie d’Eilenberg-Mazur (en anglais Eilenberg–Mazur swindle), nommée ainsi en référence à Samuel Eilenberg et Barry Mazur, est une méthode de preuve exploitant des propriétés paradoxales de sommes infinies. En topologie, elle a été introduite par Barry Mazur et est souvent appelée l’escroquerie de Mazur (Mazur swindle). En algèbre, elle a été introduite par Samuel Eilenberg et porte les noms d’escroquerie d’Eilenberg (Eilenberg swindle) ou de télescope d’Eilenberg (Eilenberg telescope) (voir aussi l’article somme télescopique).

L’escroquerie d’Eilenberg-Mazur s’apparente à la démonstration humoristique classique suivante affirmant que 1 = 0 :

1 = 1 + (-1 + 1) + (-1 + 1) + \dots = 1 - 1 + 1 - 1 + \dots = (1 - 1) + (1 - 1) + \dots = 0

Cette « preuve » n’est évidemment pas valide en tant que résultat concernant les nombres réels, car la série de Grandi 1 - 1 + 1 - 1 + \dots n’est pas convergente. Cependant, un raisonnement analogue peut être employé dans certains contextes où une forme d’« addition » appropriée existe sur des objets mathématiques, pour lesquels des sommes infinies ont effectivement un sens. Dans ces contextes, un tel argument permet de démontrer que si A + B = 0, alors nécessairement A = B = 0.

Mazur swindle

In geometric topology the addition used in the swindle is usually the connected sum of knots or manifolds.

Example (Rolfsen 1990, chapter 4B): A typical application of the Mazur swindle in geometric topology is the proof that the sum of two non-trivial knots A and B is non-trivial. For knots it is possible to take infinite sums by making the knots smaller and smaller, so if A + B is trivial then

$A=A+(B+A)+(B+A)+\cdots =(A+B)+(A+B)+\cdots =0\,$

so A is trivial (and B by a similar argument). The infinite sum of knots is usually a wild knot, not a tame knot. See (Poénaru 2007) for more geometric examples.

Example: The oriented n-manifolds have an addition operation given by connected sum, with 0 the n-sphere. If A + B is the n-sphere, then A + B + A + B + ... is Euclidean space so the Mazur swindle shows that the connected sum of A and Euclidean space is Euclidean space, which shows that A is the 1-point compactification of Euclidean space and therefore A is homeomorphic to the n-sphere. (This does not show in the case of smooth manifolds that A is diffeomorphic to the n-sphere, and in some dimensions, such as 7, there are examples of exotic spheres A with inverses that are not diffeomorphic to the standard n-sphere.)

Eilenberg swindle

In algebra the addition used in the swindle is usually the direct sum of modules over a ring.

Example: A typical application of the Eilenberg swindle in algebra is the proof that if A is a projective module over a ring R then there is a free module F with A ⊕ F ≅ F. To see this, choose a module B such that A ⊕ B is free, which can be done as A is projective, and put

F = B ⊕ A ⊕ B ⊕ A ⊕ B ⊕ ⋯.

so that

A ⊕ F = A ⊕ (B ⊕ A) ⊕ (B ⊕ A) ⊕ ⋯ = (A ⊕ B) ⊕ (A ⊕ B) ⊕ ⋯ ≅ F.

Example: (Eisenbud 1995, p.121) Finitely generated free modules over commutative rings R have a well-defined natural number as their dimension which is additive under direct sums, and are isomorphic if and only if they have the same dimension.

This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be an abelian group such that X ≅ X ⊕ X (for example the direct sum of an infinite number of copies of any nonzero abelian group), and let R be the ring of endomorphisms of X. Then the left R-module R is isomorphic to the left R-module R ⊕ R.

Example: (Lam 2003, Exercise 8.16) If A and B are any groups then the Eilenberg swindle can be used to construct a ring R such that the group rings R[A] and R[B] are isomorphic rings: take R to be the group ring of the restricted direct product of infinitely many copies of A ⨯ B.

Other examples

The proof of the Cantor–Bernstein–Schroeder theorem might be seen as antecedent of the Eilenberg–Mazur swindle. In fact, the ideas are quite similar. If there are injections of sets from X to Y and from Y to X, this means that formally we have X = Y + A and Y = X + B for some sets A and B, where + means disjoint union and = means there is a bijection between two sets. Expanding the former with the latter,

X = X + A + B.

In this bijection, let Z consist of those elements of the left hand side that correspond to an element of X on the right hand side. This bijection then expands to the bijection

X = A + B + A + B + ⋯ + Z.

Substituting the right hand side for X in Y = B + X gives the bijection

Y = B + A + B + A + ⋯ + Z.

Switching every adjacent pair B + A yields

Y = A + B + A + B + ⋯ + Z.

Composing the bijection for X with the inverse of the bijection for Y then yields

X = Y.

This argument depended on the bijections A + B = B + A and A + (B + C) = (A + B) + C as well as the well-definedness of infinite disjoint union.

References

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External links

• Exposition by Terence Tao on Mazur's swindle in topology